On constant-weight TSP-tours

Is it possible to label the edges of Kₙ with distinct integer weights so that every Hamilton cycle has the same total weight? We give a local condition characterizing the labellings that witness this question's perhaps surprising affirmative answer. More generally, we address the question that arises when "Hamilton cycle" is replaced by "k-factor" for nonnegative integers k. Such edge-labellings are in correspondence with certain vertex-labellings, and the link allows us to determine (up to a constant factor) the growth rate of the maximum edge-label in a "most efficient" injective metric trivial-TSP labelling.